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The second virial coefficient of diethyl ketone from measurements of the excess molar enthalpy of (nitrogen +diethyl ketone)(g)
J. Chem. Thermodynamics 1998, 30, 951]957 Article No. ct980361
The second virial coefficient of diethyl ketone from measurements of the excess molar enthalpy of ( nitrogen H diethyl ketone) ( g) C. Mathonat, J. Wilson, and C. J. Wormald School of Chemistry, Uni¨ ersity of Bristol, Bristol BS8 1TS, U.K.
A flow-mixing calorimeter has been used to measure the excess molar enthalpy HmE Ž p8. of Žnitrogen q diethyl ketone.Žg. at standard atmospheric pressure Ž p8. over the temperature range Ž373.2 to 423.2. K. The measurements were analysed using pair potential parameters for nitrogen, assuming suitable combining rules for the calculation of cross-terms, and, for diethyl ketone, finding parameters for the Stockmayer potential which fitted the excess enthalpy measurements. These parameters, which are «rk s 720 K, s s 0.375 nm, and t* s 0.506, yielded values of the second virial coefficient B for diethyl ketone in satisfactory agreement with other work. Q 1998 Academic Press KEYWORDS: excess enthalpy; second virial coefficient; gas mixture
1. Introduction In a previous paper Ž1. we reported measurements of the excess molar enthalpy of Žnitrogen q dimethyl ketone.Žg. made using a differential flow-mixing calorimeter. Second virial coefficients B for dimethyl ketone were derived from the measurements and these were shown to be consistent with measurements of the isothermal Joule]Thomson coefficient f extrapolated to zero pressure, where f s B y T Žd BrdT ., and the quantity x s T ŽdC prdT . s yT 2 Žd 2 BrdT 2 ., obtained from the pressure derivative of the heat capacity at the zero pressure limit. A following paper Ž2. reported measurements of the excess molar enthalpy of Žcyclohexane q dimethyl ketone.Žg. and Žbenzene q dimethyl ketone.Žg.. Analysis of the Žcyclohexane q dimethyl ketone.Žg. measurements gave information about the combining rule for the second virial cross coefficients, and this rule was then used to predict HmE for Žbenzene q dimethyl ketone.Žg.. The experimental values of HmE for Žbenzene q dimethyl ketone.Žg. were much smaller than the predicted values, and the difference was attributed to charge transfer complex formation. A quasi-chemical model was used to analyse the difference, and this yielded a value of the equilibrium constant K 12 Ž298.15 K. s 0.349 MPay1 and an enthalpy of formation of the complex D H12 s yŽ16.4 " 2. kJ . moly1 . This enthalpy of association is bigger than that for benzene]water Ž3. D H12 s yŽ12 " 2. kJ . moly1 4 , or benzene]methanol Ž4. D H12 s yŽ13 " 2. kJ . moly1 4 , or benzene]ethanol Ž5. 0021]9614r98r080951 q 07 $30.00r0
Q Academic Press
952
C. Mathonat, J. Wilson, and C. J. Wormald
D H12 s yŽ14 " 2. kJ . moly1 4 . Suzuki et al.Ž6. observed fully rotationally resolved spectra of three isotopic species of 1 : 1 clusters of benzene with water and showed that the water molecule is in free rotation above the plane of the benzene ring. This was confirmed by other groups.Ž7 ] 10. Suzuki et al. further speculated that benzene forms hydrogen bonds with water.Ž6. If this is the correct explanation it is hard to see why the benzene]dimethyl ketone interaction energy can be so large. Dimethyl ketone is not self-associated, and has no hydroxyl groups with which to form a hydrogen bond. To investigate this problem further we wished to make parallel measurements on Žbenzene q diethyl ketone., but found that the only second virial coefficients for this substance were values calculated from measurements of vapour pressures and enthalpies of vaporization at three temperatures.Ž11. It was therefore necessary to obtain further values of B for diethyl ketone, and we chose to do this in the same way as for dimethyl ketone, by measuring the excess enthalpy of Žnitrogen q diethyl ketone. vapour.
2. Experimental Most of the vapour phase excess enthalpy measurements reported previously were made using a differential flow-mixing calorimeter Ž12. which had the advantage that any Joule]Thomson effect in the calorimeter was cancelled out at source. Recently Ž13. a flow-mixing apparatus with a single calorimeter was described. A mercury manometer was connected across the calorimeter so that the pressure drop could be measured, and this allowed a correction for the Joule]Thomson effect in the calorimeter to be made. The calorimeter was tested on Žnitrogen q cyclohexane.Žg., and three strategies for making measurements were described. In the first, the baseline on the chart recorder connected to the platinum resistance thermometer in the calorimeter was obtained by passing premixed Žnitrogen q cyclohexane. gas through the apparatus. The baseline so obtained included a contribution from the Joule]Thomson effect in the calorimeter. The two pure components were then passed through the calorimeter to form the mixture, each flowing at the same rate used for the baseline determination. Under these conditions the Joule]Thomson effect obtained in the baseline determination was the same as in the mixing experiment, and it was not necessary to make any correction for the effect. In the second strategy the baseline was determined by flowing pure nitrogen through the apparatus. In the range of interest, at temperatures above 373.2 K, the Joule]Thomson coefficient of nitrogen is very small compared to that for cyclohexane, and the correction for this effect can be neglected. It is, however, necessary to correct the measurements for the Joule]Thomson effect of the mixture which flows through the calorimeter, but this is easily done. The measurements on Žnitrogen q diethyl ketone.Žg. were all made by obtaining an initial baseline on nitrogen. The mass fraction of the nitrogen was 0.9998. The diethyl ketone ŽC 2 H 5 . 2 CO was supplied at a mole fraction of 0.99 and was used as supplied. The measurements were made with pure component flow rates in the range Ž1 to 1.2. mmol . sy1 and the apparatus was adjusted to give mixtures of mole
953
Second virial coefficient of diethyl ketone
TABLE 1. The excess molar enthalpy HmE Ž p8. of 0.5N2 q 0.5ŽC 2 H 5 . 2 CO4Žg. and derived quantities at experimental temperatures T. HmE Ž B . and HmE Ž C . are, respectively, the contributions of the second and third virial coefficients to the excess molar enthalpy, B12 and f 12 are the second virial coefficient and the isothermal Joule]Thomson coefficient of the binary mixture, and B22 and f 22 are the second virial coefficient and the isothermal Joule]Thomson coefficient of diethyl ketone, respectively HmE Ž C . J . moly1
HmE Ž B . J . moly1
B12
f 12
B22
f 22
K
HmE Ž p8 . J . moly1
cm3 . moly1
cm3 . moly1
cm3 . moly1
cm3 . moly1
373.2 379.2 383.2 393.2 403.2 413.2 423.2
166.4 154.0 145.1 129.8 115.3 103.5 94.5
2.91 2.42 2.05 1.44 1.00 0.69 0.46
163.5 151.6 143.0 128.4 114.3 102.8 94.0
y90 y87 y85 y81 y77 y73 y70
y268 y261 y256 y246 y235 y226 y217
y1449 y1373 y1326 y1219 y1125 y1043 y971
y6406 y5981 y5721 y5142 y4648 y4224 y3858
T
fraction y in the range y s 0.47 to y s 0.53. As the normal boiling temperature of diethyl ketone is T s 374.8 K the measurements at T s 373.15 K were made at pressures of around 90 kPa. All other measurements were made at ambient atmospheric pressure. It was a simple matter to adjust the HmE measurements to y s 0.5 and p8 s 0.101325 MPa. Usually five runs were performed at each experimental temperature, and these were averaged to give the results listed in table 1.
3. Joule–Thomson corrections and analysis of HmE measurements The excess molar enthalpy of a binary gaseous mixture can be writtenŽ1. as the sum of terms which are a function of the second virial coefficient B, and the third virial coefficient C, etc. HmE s HmE Ž B . q HmE Ž C . q ??? HmE Ž B .
Ž 1.
can be written as: HmE Ž B . s y Ž 1 y y . p Ž 2 f 12 y f 11 y f 22 . y
Ž p 2rRT . Bf y yB11 f 11 y Ž 1 y y . B22 f 22 4 ,
Ž 2.
where y is the mole fraction of component 1 and the isothermal Joule]Thomson coefficient f is given by, f s B y T Ž d BrdT . . Ž 3. The quantity B without a subscript refers to the mixture and is given by: 2
B s y 2 B11 q 2 y Ž 1 y y . B12 q Ž 1 y y . B22 , and f is given by a similar equation. HmE
HmE Ž C .
Ž 4.
can be written:
Ž C . s Ž p rRT . c y yc 111 y Ž 1 y y . f 222 4 , 2
Ž 5.
where,
c s C y Ž Tr2 . Ž dCrdT . .
Ž 6.
954
C. Mathonat, J. Wilson, and C. J. Wormald
In equation Ž5. c without a subscript is for the mixture and is given by: 2
3
c s y 3c 111 q 3 y 2 Ž 1 y y . c 112 q 3 y Ž 1 y y . c 122 q Ž 1 y y . c 222 .
Ž 7.
To analyse our measurements we first make our best estimate of HmE Ž C . and subtract it from the measured values of HmE to yield the quantity HmE Ž B .. As shown previously Ž13. HmE is given by: HmE s VIrf y f pŽmixture. D p y Ž 1r2 . f p Ž N2 . D p,
Ž 8.
where the VI Žvoltage = current . product is the power flowing through the calorimeter heater when isothermal conditions have been obtained, f is the molar flow rate of the mixture, and D p is the pressure drop across the calorimeter. The quantity f p is given by:
f p s f Ž 1 y 2 pBrRT . ,
Ž 9.
where B and f are the second virial coefficient and the isothermal Joule]Thomson coefficient of the mixture, respectively. Under the conditions of the experiment the Joule]Thomson coefficient of nitrogen is very small, and the right-hand term in equation Ž8. can be neglected. To analyse our measurements we used the Kihara spherical core potential for nitrogen, and the Stockmayer potential for diethyl ketone. We assumed arithmetic mean combining rules to obtain the cross parameters a12 and s 12 , and for « 12 we used the geometric mean rule:
« 12 s Ž 1 y k 12 . Ž « 11 « 22 .
1r2
.
Ž 10 .
For the Žnitrogen q diethyl ketone. mixture the cross-terms B12 and f 12 are small compared with B22 and f 22 for diethyl ketone, and it makes little difference to the calculations if we calculate « 12 using the combining rule of Hudson and McCoubrey,Ž14. which may be inappropriate anyway for a mixture with such a highly polar component, or simply set Ž1 y k 12 . s 1, which is what we chose to do. To obtain potential parameters for diethyl ketone an iterative procedure was used, and this was begun by finding the values of f 22 and f 12 which balanced the equation: VIrf s y Ž 1 y y . p Ž 2 f 12 y f 11 y f 22 . .
Ž 11 .
To calculate B11 and f 11 for nitrogen we used the Kihara potential with the parameters «rk s 139.2 K, s s 0.3526 nm, and a s 0.03526 nm fitted by Tee et al.Ž15. The equation was balanced by creating a two-dimensional grid of possible values of the Stockmayer parameters «rk and s for diethyl ketone. For each pair of values f 22 and f 12 at each experimental temperature were calculated, hence all quantities on the right-hand side of equation Ž11. were known. The pair of values which gave best overall agreement with the set of experimental quantities VIrf was identified, and were used to calculate the p 2 term in equation Ž2. and the
Second virial coefficient of diethyl ketone
955
FIGURE 1. The contribution HmE Ž B . to the excess molar enthalpy of 0.5N2 q 0.5ŽC 2 H 5 . 2 CO4 arising from second virial coefficient terms only. `, values from table 1; }}}, calculated from equation Ž2. using the potential parameters given in the text.
Joule]Thomson correction term f pŽmixture. in equation Ž8.. These terms were subtracted from VIrf and the grid search method was again used to find new values of «rk and s from which f 22 and f 12 were calculated, and which balanced equation Ž11.. After five cycles of this iterative procedure, reducing the size of the increments between points on the grid each time so as to improve the accuracy with which the potential parameters could be determined, constant values of «rk and s were obtained. These parameters, «rk s 720 K and s s 0.375 nm, combined with the dipole moment m s 9.14 . 10y3 0 Cm†, lead to a value of the Stockmayer reduced dipole parameter t* s 0.506. This reduced dipole moment is calculated from the formula t* s m 2r 8 1r2 . Ž «rk . . s 3 4 . The value of t* for diethyl ketone is about half as big as that for dimethyl ketone Ž t* s 0.956.; the ratio is actually 0.53 : 1. We had expected the value of t* for diethyl ketone to be much closer to that for dimethyl ketone, and were concerned that it was wrong. Some reassurance was gained by calculating the ratio of the quantities t c s m2rŽTc Vc . for the two ketones, which is 0.51 : 1. The parameters «rk s 720, s s 0.375 nm, and t* s 0.506 were used to calculate final values of B22 and f 22 . The results of this analysis are listed in table 1 and the values of HmE Ž B . are plotted in figure 1. † Editorial comment: molecular dipole moments are often expressed in the non-SI unit debye D, where D f 3.33564 . 10y3 0 Cm.
956
C. Mathonat, J. Wilson, and C. J. Wormald
FIGURE 2. The second virial coefficient B of diethyl ketone. `, values from Table 1; ^ reference Ž11.; }}}, calculated from the Stockmayer potential using the parameters «rk s 720 K, s s 0.375 nm, and t* s 0.506.
4. Second virial coefficients for diethyl ketone In their compilation of virial coefficients Dymond and SmithŽ16. list only three values of B for diethyl ketone, and these were determined indirectly from measurements of the enthalpy of vaporization and the vapour pressure.Ž11. The uncertainty on these values is between "Ž50 and 100. cm3 . moly1 . Figure 2 shows the reference 12 virial coefficients compared with those calculated from the above Stockmayer parameters. Virial coefficients obtained from Ž p,V,T . measurements are often more negative than those obtained using flow calorimetric techniques, particularly at low temperatures, and the difference is usually ascribed to adsorption errors in the Ž p, V, T . apparatus. However, there is no reason why the values of B calculated from vapour pressures and enthalpies of vaporization should be subject to this source of error. Values of B calculated this way are obtained by taking the difference between two large quantities, and a small error in one of these can introduce a large error into B. Values of B22 and f 22 for diethyl ketone are listed in table 1. The uncertainty in the values of B22 is estimated to be "40 cm3 . moly1 , and that in f 22 is estimated to be "100 cm3 . moly1 . Also listed in table 1 are the values of B12 and f 12 . Compared with B22 and f 22 all these cross coefficients are small, and are less than 7 per cent of the corresponding values for diethyl ketone. This reinforces the point made earlier, that the analysis is insensitive to the choice of combining rule for « 12 . Agreement between the two sets of measurements shown in figure 2 is to within the combined experimental error. REFERENCES 1. Doyle, J; Hutchings, D. J.; Lancaster N. M.; Wormald C. J. J. Chem. Thermodynamics 1997, 29, 677]685.
Second virial coefficient of diethyl ketone 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
957
Wormald, C. J.; Mayr J. C. J. Chem. Soc. Faraday Trans. 1998, 94, 207]212. Wormald, C. J.; Lancaster, N. M.; Sowden; C. J. J. Chem. Soc. Faraday Trans. 1997, 93, 1921]1926. Wormald, C. J.; Sowden, C. J. Int. J. Thermophysics 1997, 18, 1465]1481. Wormald, C. J.; Sowden, C. J. J. Chem. Thermodynamics 1997, 29, 1223]1236. Suzuki, S; Green, P. G.; Bumgarner, R. E.; Dasgupta, S.; Goddard, W. A.; Blake, G. A. Science 1992, 257, 942]945. Gotch, A. J.; Zweir, T. S. J. Chem. Phys. 1992, 96, 3388]3401. Augsberger, J. D.; Dykstra, C. E. J. Phys. Chem. 1993, 97, 980]984. Gutowsky, H. S.; Emilsson, T.; Arunan, E. J. Phys. Chem. 1995, 103, 3917]3925. Garret, A. W.; Severance, D. L.; Zweir, T. S. J. Chem. Phys. 1992, 96, 7245]7258. Hales, J. L.; Lees, E. B.; Ruxton, D. J. Trans. Faraday Soc. 1967, 63, 1876]1886. Wormald, C. J. J. Chem. Thermodynamics 1977, 9, 901]910. Wormald, C. J. J. Chem. Thermodynamics 1997, 29, 701]714. Hudson, G. H.; McCoubrey, J. C. Trans. Faraday Soc. 1960, 56, 761]766. Tee, L. S.; Gotoh, S.; Stewart, W. E. Ind. Eng. Chem. Fundamentals 1966, 5, 363]366. Dymond, J. H.; Smith. E. B. The Virial Coefficients of Pure Gases and Mixtures. Clarendon Press: Oxford. 1980.
(Recei¨ ed 20 August 1997; in final form 17 February 1998)
WA97r057
The second virial coefficient of diethyl ketone from measurements of the excess molar enthalpy of ( nitrogen H diethyl ketone) ( g) C. Mathonat, J. Wilson, and C. J. Wormald School of Chemistry, Uni¨ ersity of Bristol, Bristol BS8 1TS, U.K.
A flow-mixing calorimeter has been used to measure the excess molar enthalpy HmE Ž p8. of Žnitrogen q diethyl ketone.Žg. at standard atmospheric pressure Ž p8. over the temperature range Ž373.2 to 423.2. K. The measurements were analysed using pair potential parameters for nitrogen, assuming suitable combining rules for the calculation of cross-terms, and, for diethyl ketone, finding parameters for the Stockmayer potential which fitted the excess enthalpy measurements. These parameters, which are «rk s 720 K, s s 0.375 nm, and t* s 0.506, yielded values of the second virial coefficient B for diethyl ketone in satisfactory agreement with other work. Q 1998 Academic Press KEYWORDS: excess enthalpy; second virial coefficient; gas mixture
1. Introduction In a previous paper Ž1. we reported measurements of the excess molar enthalpy of Žnitrogen q dimethyl ketone.Žg. made using a differential flow-mixing calorimeter. Second virial coefficients B for dimethyl ketone were derived from the measurements and these were shown to be consistent with measurements of the isothermal Joule]Thomson coefficient f extrapolated to zero pressure, where f s B y T Žd BrdT ., and the quantity x s T ŽdC prdT . s yT 2 Žd 2 BrdT 2 ., obtained from the pressure derivative of the heat capacity at the zero pressure limit. A following paper Ž2. reported measurements of the excess molar enthalpy of Žcyclohexane q dimethyl ketone.Žg. and Žbenzene q dimethyl ketone.Žg.. Analysis of the Žcyclohexane q dimethyl ketone.Žg. measurements gave information about the combining rule for the second virial cross coefficients, and this rule was then used to predict HmE for Žbenzene q dimethyl ketone.Žg.. The experimental values of HmE for Žbenzene q dimethyl ketone.Žg. were much smaller than the predicted values, and the difference was attributed to charge transfer complex formation. A quasi-chemical model was used to analyse the difference, and this yielded a value of the equilibrium constant K 12 Ž298.15 K. s 0.349 MPay1 and an enthalpy of formation of the complex D H12 s yŽ16.4 " 2. kJ . moly1 . This enthalpy of association is bigger than that for benzene]water Ž3. D H12 s yŽ12 " 2. kJ . moly1 4 , or benzene]methanol Ž4. D H12 s yŽ13 " 2. kJ . moly1 4 , or benzene]ethanol Ž5. 0021]9614r98r080951 q 07 $30.00r0
Q Academic Press
952
C. Mathonat, J. Wilson, and C. J. Wormald
D H12 s yŽ14 " 2. kJ . moly1 4 . Suzuki et al.Ž6. observed fully rotationally resolved spectra of three isotopic species of 1 : 1 clusters of benzene with water and showed that the water molecule is in free rotation above the plane of the benzene ring. This was confirmed by other groups.Ž7 ] 10. Suzuki et al. further speculated that benzene forms hydrogen bonds with water.Ž6. If this is the correct explanation it is hard to see why the benzene]dimethyl ketone interaction energy can be so large. Dimethyl ketone is not self-associated, and has no hydroxyl groups with which to form a hydrogen bond. To investigate this problem further we wished to make parallel measurements on Žbenzene q diethyl ketone., but found that the only second virial coefficients for this substance were values calculated from measurements of vapour pressures and enthalpies of vaporization at three temperatures.Ž11. It was therefore necessary to obtain further values of B for diethyl ketone, and we chose to do this in the same way as for dimethyl ketone, by measuring the excess enthalpy of Žnitrogen q diethyl ketone. vapour.
2. Experimental Most of the vapour phase excess enthalpy measurements reported previously were made using a differential flow-mixing calorimeter Ž12. which had the advantage that any Joule]Thomson effect in the calorimeter was cancelled out at source. Recently Ž13. a flow-mixing apparatus with a single calorimeter was described. A mercury manometer was connected across the calorimeter so that the pressure drop could be measured, and this allowed a correction for the Joule]Thomson effect in the calorimeter to be made. The calorimeter was tested on Žnitrogen q cyclohexane.Žg., and three strategies for making measurements were described. In the first, the baseline on the chart recorder connected to the platinum resistance thermometer in the calorimeter was obtained by passing premixed Žnitrogen q cyclohexane. gas through the apparatus. The baseline so obtained included a contribution from the Joule]Thomson effect in the calorimeter. The two pure components were then passed through the calorimeter to form the mixture, each flowing at the same rate used for the baseline determination. Under these conditions the Joule]Thomson effect obtained in the baseline determination was the same as in the mixing experiment, and it was not necessary to make any correction for the effect. In the second strategy the baseline was determined by flowing pure nitrogen through the apparatus. In the range of interest, at temperatures above 373.2 K, the Joule]Thomson coefficient of nitrogen is very small compared to that for cyclohexane, and the correction for this effect can be neglected. It is, however, necessary to correct the measurements for the Joule]Thomson effect of the mixture which flows through the calorimeter, but this is easily done. The measurements on Žnitrogen q diethyl ketone.Žg. were all made by obtaining an initial baseline on nitrogen. The mass fraction of the nitrogen was 0.9998. The diethyl ketone ŽC 2 H 5 . 2 CO was supplied at a mole fraction of 0.99 and was used as supplied. The measurements were made with pure component flow rates in the range Ž1 to 1.2. mmol . sy1 and the apparatus was adjusted to give mixtures of mole
953
Second virial coefficient of diethyl ketone
TABLE 1. The excess molar enthalpy HmE Ž p8. of 0.5N2 q 0.5ŽC 2 H 5 . 2 CO4Žg. and derived quantities at experimental temperatures T. HmE Ž B . and HmE Ž C . are, respectively, the contributions of the second and third virial coefficients to the excess molar enthalpy, B12 and f 12 are the second virial coefficient and the isothermal Joule]Thomson coefficient of the binary mixture, and B22 and f 22 are the second virial coefficient and the isothermal Joule]Thomson coefficient of diethyl ketone, respectively HmE Ž C . J . moly1
HmE Ž B . J . moly1
B12
f 12
B22
f 22
K
HmE Ž p8 . J . moly1
cm3 . moly1
cm3 . moly1
cm3 . moly1
cm3 . moly1
373.2 379.2 383.2 393.2 403.2 413.2 423.2
166.4 154.0 145.1 129.8 115.3 103.5 94.5
2.91 2.42 2.05 1.44 1.00 0.69 0.46
163.5 151.6 143.0 128.4 114.3 102.8 94.0
y90 y87 y85 y81 y77 y73 y70
y268 y261 y256 y246 y235 y226 y217
y1449 y1373 y1326 y1219 y1125 y1043 y971
y6406 y5981 y5721 y5142 y4648 y4224 y3858
T
fraction y in the range y s 0.47 to y s 0.53. As the normal boiling temperature of diethyl ketone is T s 374.8 K the measurements at T s 373.15 K were made at pressures of around 90 kPa. All other measurements were made at ambient atmospheric pressure. It was a simple matter to adjust the HmE measurements to y s 0.5 and p8 s 0.101325 MPa. Usually five runs were performed at each experimental temperature, and these were averaged to give the results listed in table 1.
3. Joule–Thomson corrections and analysis of HmE measurements The excess molar enthalpy of a binary gaseous mixture can be writtenŽ1. as the sum of terms which are a function of the second virial coefficient B, and the third virial coefficient C, etc. HmE s HmE Ž B . q HmE Ž C . q ??? HmE Ž B .
Ž 1.
can be written as: HmE Ž B . s y Ž 1 y y . p Ž 2 f 12 y f 11 y f 22 . y
Ž p 2rRT . Bf y yB11 f 11 y Ž 1 y y . B22 f 22 4 ,
Ž 2.
where y is the mole fraction of component 1 and the isothermal Joule]Thomson coefficient f is given by, f s B y T Ž d BrdT . . Ž 3. The quantity B without a subscript refers to the mixture and is given by: 2
B s y 2 B11 q 2 y Ž 1 y y . B12 q Ž 1 y y . B22 , and f is given by a similar equation. HmE
HmE Ž C .
Ž 4.
can be written:
Ž C . s Ž p rRT . c y yc 111 y Ž 1 y y . f 222 4 , 2
Ž 5.
where,
c s C y Ž Tr2 . Ž dCrdT . .
Ž 6.
954
C. Mathonat, J. Wilson, and C. J. Wormald
In equation Ž5. c without a subscript is for the mixture and is given by: 2
3
c s y 3c 111 q 3 y 2 Ž 1 y y . c 112 q 3 y Ž 1 y y . c 122 q Ž 1 y y . c 222 .
Ž 7.
To analyse our measurements we first make our best estimate of HmE Ž C . and subtract it from the measured values of HmE to yield the quantity HmE Ž B .. As shown previously Ž13. HmE is given by: HmE s VIrf y f pŽmixture. D p y Ž 1r2 . f p Ž N2 . D p,
Ž 8.
where the VI Žvoltage = current . product is the power flowing through the calorimeter heater when isothermal conditions have been obtained, f is the molar flow rate of the mixture, and D p is the pressure drop across the calorimeter. The quantity f p is given by:
f p s f Ž 1 y 2 pBrRT . ,
Ž 9.
where B and f are the second virial coefficient and the isothermal Joule]Thomson coefficient of the mixture, respectively. Under the conditions of the experiment the Joule]Thomson coefficient of nitrogen is very small, and the right-hand term in equation Ž8. can be neglected. To analyse our measurements we used the Kihara spherical core potential for nitrogen, and the Stockmayer potential for diethyl ketone. We assumed arithmetic mean combining rules to obtain the cross parameters a12 and s 12 , and for « 12 we used the geometric mean rule:
« 12 s Ž 1 y k 12 . Ž « 11 « 22 .
1r2
.
Ž 10 .
For the Žnitrogen q diethyl ketone. mixture the cross-terms B12 and f 12 are small compared with B22 and f 22 for diethyl ketone, and it makes little difference to the calculations if we calculate « 12 using the combining rule of Hudson and McCoubrey,Ž14. which may be inappropriate anyway for a mixture with such a highly polar component, or simply set Ž1 y k 12 . s 1, which is what we chose to do. To obtain potential parameters for diethyl ketone an iterative procedure was used, and this was begun by finding the values of f 22 and f 12 which balanced the equation: VIrf s y Ž 1 y y . p Ž 2 f 12 y f 11 y f 22 . .
Ž 11 .
To calculate B11 and f 11 for nitrogen we used the Kihara potential with the parameters «rk s 139.2 K, s s 0.3526 nm, and a s 0.03526 nm fitted by Tee et al.Ž15. The equation was balanced by creating a two-dimensional grid of possible values of the Stockmayer parameters «rk and s for diethyl ketone. For each pair of values f 22 and f 12 at each experimental temperature were calculated, hence all quantities on the right-hand side of equation Ž11. were known. The pair of values which gave best overall agreement with the set of experimental quantities VIrf was identified, and were used to calculate the p 2 term in equation Ž2. and the
Second virial coefficient of diethyl ketone
955
FIGURE 1. The contribution HmE Ž B . to the excess molar enthalpy of 0.5N2 q 0.5ŽC 2 H 5 . 2 CO4 arising from second virial coefficient terms only. `, values from table 1; }}}, calculated from equation Ž2. using the potential parameters given in the text.
Joule]Thomson correction term f pŽmixture. in equation Ž8.. These terms were subtracted from VIrf and the grid search method was again used to find new values of «rk and s from which f 22 and f 12 were calculated, and which balanced equation Ž11.. After five cycles of this iterative procedure, reducing the size of the increments between points on the grid each time so as to improve the accuracy with which the potential parameters could be determined, constant values of «rk and s were obtained. These parameters, «rk s 720 K and s s 0.375 nm, combined with the dipole moment m s 9.14 . 10y3 0 Cm†, lead to a value of the Stockmayer reduced dipole parameter t* s 0.506. This reduced dipole moment is calculated from the formula t* s m 2r 8 1r2 . Ž «rk . . s 3 4 . The value of t* for diethyl ketone is about half as big as that for dimethyl ketone Ž t* s 0.956.; the ratio is actually 0.53 : 1. We had expected the value of t* for diethyl ketone to be much closer to that for dimethyl ketone, and were concerned that it was wrong. Some reassurance was gained by calculating the ratio of the quantities t c s m2rŽTc Vc . for the two ketones, which is 0.51 : 1. The parameters «rk s 720, s s 0.375 nm, and t* s 0.506 were used to calculate final values of B22 and f 22 . The results of this analysis are listed in table 1 and the values of HmE Ž B . are plotted in figure 1. † Editorial comment: molecular dipole moments are often expressed in the non-SI unit debye D, where D f 3.33564 . 10y3 0 Cm.
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C. Mathonat, J. Wilson, and C. J. Wormald
FIGURE 2. The second virial coefficient B of diethyl ketone. `, values from Table 1; ^ reference Ž11.; }}}, calculated from the Stockmayer potential using the parameters «rk s 720 K, s s 0.375 nm, and t* s 0.506.
4. Second virial coefficients for diethyl ketone In their compilation of virial coefficients Dymond and SmithŽ16. list only three values of B for diethyl ketone, and these were determined indirectly from measurements of the enthalpy of vaporization and the vapour pressure.Ž11. The uncertainty on these values is between "Ž50 and 100. cm3 . moly1 . Figure 2 shows the reference 12 virial coefficients compared with those calculated from the above Stockmayer parameters. Virial coefficients obtained from Ž p,V,T . measurements are often more negative than those obtained using flow calorimetric techniques, particularly at low temperatures, and the difference is usually ascribed to adsorption errors in the Ž p, V, T . apparatus. However, there is no reason why the values of B calculated from vapour pressures and enthalpies of vaporization should be subject to this source of error. Values of B calculated this way are obtained by taking the difference between two large quantities, and a small error in one of these can introduce a large error into B. Values of B22 and f 22 for diethyl ketone are listed in table 1. The uncertainty in the values of B22 is estimated to be "40 cm3 . moly1 , and that in f 22 is estimated to be "100 cm3 . moly1 . Also listed in table 1 are the values of B12 and f 12 . Compared with B22 and f 22 all these cross coefficients are small, and are less than 7 per cent of the corresponding values for diethyl ketone. This reinforces the point made earlier, that the analysis is insensitive to the choice of combining rule for « 12 . Agreement between the two sets of measurements shown in figure 2 is to within the combined experimental error. REFERENCES 1. Doyle, J; Hutchings, D. J.; Lancaster N. M.; Wormald C. J. J. Chem. Thermodynamics 1997, 29, 677]685.
Second virial coefficient of diethyl ketone 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
957
Wormald, C. J.; Mayr J. C. J. Chem. Soc. Faraday Trans. 1998, 94, 207]212. Wormald, C. J.; Lancaster, N. M.; Sowden; C. J. J. Chem. Soc. Faraday Trans. 1997, 93, 1921]1926. Wormald, C. J.; Sowden, C. J. Int. J. Thermophysics 1997, 18, 1465]1481. Wormald, C. J.; Sowden, C. J. J. Chem. Thermodynamics 1997, 29, 1223]1236. Suzuki, S; Green, P. G.; Bumgarner, R. E.; Dasgupta, S.; Goddard, W. A.; Blake, G. A. Science 1992, 257, 942]945. Gotch, A. J.; Zweir, T. S. J. Chem. Phys. 1992, 96, 3388]3401. Augsberger, J. D.; Dykstra, C. E. J. Phys. Chem. 1993, 97, 980]984. Gutowsky, H. S.; Emilsson, T.; Arunan, E. J. Phys. Chem. 1995, 103, 3917]3925. Garret, A. W.; Severance, D. L.; Zweir, T. S. J. Chem. Phys. 1992, 96, 7245]7258. Hales, J. L.; Lees, E. B.; Ruxton, D. J. Trans. Faraday Soc. 1967, 63, 1876]1886. Wormald, C. J. J. Chem. Thermodynamics 1977, 9, 901]910. Wormald, C. J. J. Chem. Thermodynamics 1997, 29, 701]714. Hudson, G. H.; McCoubrey, J. C. Trans. Faraday Soc. 1960, 56, 761]766. Tee, L. S.; Gotoh, S.; Stewart, W. E. Ind. Eng. Chem. Fundamentals 1966, 5, 363]366. Dymond, J. H.; Smith. E. B. The Virial Coefficients of Pure Gases and Mixtures. Clarendon Press: Oxford. 1980.
(Recei¨ ed 20 August 1997; in final form 17 February 1998)
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